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Modal Science
Timothy Williamson
Current studies of the metaphysics and epistemology of modality tends to focus on the metaphysics
and epistemology of folk modal beliefs. However, their proper subject matter also includes the
metaphysics and epistemology of scientific modal beliefs — if any, but it is fairly clear that there are
many. Of course, just as scientific theorizing in general has developed out of folk thinking, so modal
scientific theorizing more specifically has developed out of modal folk thinking. But the emphasis
here will be on the scientific theorizing. (This is not to deny that folk modal beliefs, e.g. those based
on imaginative exercises, often constitute modal knowledge.)
The modalities to be discussed will be alethic, not epistemic or doxastic; they concern how things
could have been, not what is consistent with what some real or ideal subject knows or believes.
The most obvious example of scientific inquiry into modality is (quantified) modal logic, at least
when it is pursued with a suitable reading of the modal operators. I have argued in Modal Logic as
Metaphysics that the appropriate methodology for such an inquiry is a broadly abductive one, in line
with that for the rest of science. This talk, however, will emphasize the modal epistemology relevant
to more mainstream natural science, including applied mathematics.
An apparent obstacle to this enterprise is the non-modal nature of the language of mathematics,
which also provides the language for much core natural science. There are no modal operators in
mathematical notation.
An obvious response is that even if mathematical and natural scientific theories are amodal in
content, true mathematical theories may still in fact be metaphysically necessary, and true natural
scientific theories at least nomically necessary. The first part of the talk will develop this response in
more detail, emphasizing that such modal glosses on scientific theories are no mere exogenous
metaphysicians’ commentary on the theories, but instead reflect the endogenous needs of the
theories given their intended applications. (If the modal aspects were ‘epiphenomenal’, how would
postulating them gain abductive support from the explanatory successes of the theory?)
The later part of the talk goes further, discussing some examples of the alethic modal dimension
internal to the content of scientific theories.
In the end, the abductive nature of scientific methodology takes care of the abductive dimension
without the need to give it exceptional treatment.
As a special case, one modal logic may comport better with natural scientific theorizing than
another. Examples are suggested below concerning the dispute between necessitism (which asserts
that necessarily everything is necessarily something) and contingentism (which denies it). Thus even
for the specifically logical study of modality, there are not just broad methodological analogies with
natural science: there are far more specific interactions.
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Scientific theories support counterfactuals
Even if a theory is cast in non-modal terms, its applications may involve the assumption that it
supports subjunctive conditionals. Of course, when a theory is applied, we may sometimes find it
natural to articulate what we take to be its consequences as subjunctive conditionals when all we
really need are indicative ones. But often one can show that the corresponding indicative conditional
would not make an adequate substitute.
Example: Causation
We wonder whether there is a causal relationship between the variable X, which we set at time t,
and the variable Y, which we observe at time t+1.
Case 1: We set X = 1 and observe Y = 0. We calculate: Y = 1─X. We have no idea whether Y = 0
because X = 1 (causally).
Case 2: We deduce from a well-confirmed scientific theory: Y = 1─X. We set X = 1 and observe Y = 0.
We reasonably conclude that Y = 0 because X = 1 (causally).
In both cases we have the material conditional X ≠ 1  Y ≠ 0. Only in the second case do we
plausibly have the subjunctive (counterfactual) conditional X ≠ 1 > Y ≠ 0.
It might be doubted whether we can assert the indicative conditional ‘If X ≠ 1 then Y ≠ 0’ since even
if we are wrong about how we set X, we are probably still right in our observation of Y. To fix this,
consider variant cases in which we don’t observe Y directly but an alarm happens to be set to go off
if Y = 1─X. The alarm goes off in both cases. Then we can assert the indicative conditional in both
cases, but the difference between them remains.
NB:
Similar comments apply under reasonable variations of the antecedent and consequent of the
conditionals.
Assume that the well-confirmed scientific theory in case 2 is cast in mathematical form and does not
use causal terms.
We are not assuming that causation can be reductively analysed in counterfactual terms, just that
counterfactuals can provide good evidence for causal claims.
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Analysis
T is a theory in a non-modal language L with at least the truth-functors ¬ (negation) and  (the
material conditional) obeying the usual classical laws; α and β may be any formulas in L.
Even if T entails only formulas of L, T may support formulas of a modal extension L+ of L. We assume
that what T supports is closed under entailment. (NB: Only single-premise closure is used below, so
support can be probabilistic.)
L+ includes the subjunctive conditional >, and dual operators □ and ◊ for metaphysical necessity and
possibility and □N and ◊N for nomic necessity and possibility. We assume, plausibly, that ¬α > α
entails □α and that ◊N… entails ◊….
Proposition: (1) implies (2):
(1) Whenever T entails α  β, T supports α > β.
(2) Whenever T entails α, T supports □α.
Proof: Suppose (1) and that T entails α. But α entails ¬α  α, so T entails ¬α  α, so by (1) T supports
¬α > α, which as above entails □α, so T supports □α.
Objection: (2) seems too strong; don’t theories in natural science support claims of nomic rather
than metaphysical necessity? One shouldn’t expect (1) to hold if, on T, α is nomically impossible.
Reply: We can finesse that point by deriving a modified result.
Proposition: (1*) implies (2*):
(1*) Whenever T entails α  β, T supports ◊Nα  (α > β).
(2*) Whenever T entails α, T supports □Nα.
Proof: Suppose (1*) and that T entails α. Since T entails ¬α  α, T supports ◊N¬α  (¬α > α) by (1*).
But as above ◊N¬α entails ◊¬α, which is inconsistent with ¬α > α since the latter entails □α as above;
hence ◊N¬α  (¬α > α) entails ¬◊N¬α, which entails □Nα. Thus T supports □Nα.
In such cases, the modal dimension is still external to the theory T, in the sense that the language of
T is non-modal. In what follows, we consider cases where the modal dimension is internal to the
scientific theory, because the language of the theory is implicitly modal.
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Alethic probability
Our concern is with the role of probabilities in science on alethic, so non-doxastic and non-epistemic,
interpretations of probability. We are also not concerned with purely frequentist interpretations,
since they lack the modal dimension: but purely frequentist interpretations are now generally
recognized as inadequate for most purposes, since even actual long-run frequencies may be
wayward and are in any case unexplanatory. For present purposes, we do not count purely
frequentist interpretations as alethic.
Alethic probability has a modal dimension because it concerns distributions over a space of genuine
possibilities, mutually exclusive and jointly exhaustive; all of them but one are counterfactual. Thus
any event with non-zero alethic probability is possible in a corresponding alethic sense.
The most celebrated role of apparently alethic probabilities in natural science is in quantum
mechanics. However, given the difficulties of interpreting quantum mechanics, it is helpful to
consider less obscure cases.
A simple example
Explanandum: When tossed 1000 times, the coin came up heads approximately 500 times.
Explanans: The coin was fair and the tosses independent.
The explanans gives a reasonable explanation of the explanandum, even if it is not the most
fundamental explanation. Probability enters the explanation in two ways. First, the explanans itself
is implicitly probabilistic: the coin is fair in the sense that the probability of heads on a given toss is
½, and the tosses are independent in the sense that the unconditional probability of an outcome of a
given toss equals its probability conditional on given outcomes of other tosses. Second, the
connection between the explanans and the explanandum is also probabilistic, since the explanans
does not entail the explanandum (it is consistent with the coin’s coming up heads every one of the
1000 times) but instead only makes the explanandum probable (in the same sense). The relevant
probabilities are not doxastic or epistemic, since the degrees of belief or evidential support of an
actual or ideal subject are quite extraneous to the explanandum and should not figure in the
explanans. Nor are the probabilities purely frequentist, since actual frequencies would not be
explanatory. But it does not follow that the explanans entails indeterminism. We should not assume
that the only alethic probabilities involve the most full-blooded sort of objective chance; they may
be more statistical in nature (a claim which itself requires much more elucidation).
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Probability and the individuation of possibilities
Necessitists and contingentists individuate possibilities differently. For a necessitist, there are merely
possible coins C1 and C2; the possibility that C1 comes up heads and C2 tails is distinct from the
possibility that C1 comes up tails and C2 heads. For a contingentist like Stalnaker in Mere
Possibilities, there may just be the single possibility that one coin comes up heads and one coin
comes up tails, because there are no such possible coins to separate the possibilities. The
contingentist may have difficulty explaining why our relevant conditional probability for that
outcome should be 1/2 rather than 1/3 (the probability it gets if outcomes are treated as
equiprobable). This is related to the choice between Fermi-Dirac and Bose-Einstein statistics (the
two sorts of statistics are needed for different purposes. For a related issue see B. Kment,
“Haecceitism, chance, and counterfactuals”, Philosophical Review 121 (2012), pp. 573-609. In
general, different ways of individuating possibilities motivate different assignments of probability,
and thereby have knock-on effects in natural science.
Even outside the quantum domain, there are thorny questions about the interpretation of alethic
probabilities. We therefore turn to a somewhat more straightforward case.
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Case study: dynamical systems and state spaces
Examples of intended applications of the theory of dynamical systems: a pendulum; the solar
system; population growth; the weather. See S. Strogatz, Nonlinear Dynamics and Chaos: With
Applications to Physics, Biology, Chemistry, and Engineering (Westview).
Mathematically, a dynamical system consists of a set S on which some geometrical or topological
structure is defined, a set T (usually either the set of real numbers or the set of integers) with an
additive structure, and a family of functions {ft}t  T indexed by T, obeying the following constraints
for all s  S and t,t*  T:
(i)
(ii)
f0(s) = s
ft(ft*(s)) = ft+t*(s)
Informally: S is the set of instantaneous states of the system, T is the set of directed lengths of time
(+1 second and ─1 second are distinct directed lengths whose sum is 0 seconds), and ft(s) is the state
of the system a length of time t after an instant when its state was s.
The states in S are maximally specific in relevant respects, mutually exclusive, and jointly exhaustive.
T includes negative as well as positive lengths of time, so the system is deterministic both forwards
and backwards in time. This interpretation verifies (i) and (ii). (But for some purposes we could
require directions to be non-negative, which would yield only forwards determinism.)
Continuous systems: If T is the set of real numbers, the functions ft are typically continuous (for the
designated topology on S) and implicitly defined by differential equations; they form a flow.
Discrete systems: If T is the set of integers, the functions ft are typically implicitly defined by
difference equations; they may still be continuous (for the designated topology on S).
The mathematics of dynamical systems is regular, non-modal mathematics.
However, most intended interpretation of the mathematics of dynamical systems are modal.
A trajectory is a set of the form {ft(s): t  T} for some s  S, i.e. the set of states the system goes
through at some time or other if at some time it is in state s.
If any trajectory exhausts S (i.e. goes through every state in S), every trajectory exhausts S. Typically,
no trajectory exhausts S. Thus the states in S are possible states, not all of which are ever actualized.
States in S are like possible worlds. Being instantaneous, they are even more like points in a Kripke
model of modal temporal logic. However, unlike such points, they are repeatable, when the system
exhibits cyclic behaviour. (NB The model theory even of modal logic is part of non-modal
mathematics.) But states are no less modal for also being temporal.
Thus standard applications of the mathematics of dynamical systems in natural science are
relevantly like reasoning about modality in terms of possible worlds.
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Example: attractors
The notion of an attractor plays an important role in the theory of dynamical systems, in helping
understand their long-term qualitative behaviour. Not all authors define the term in exactly
equivalent ways, but the following definition is fairly standard (see Strogatz, p. 324). It assumes a
topological structure on S.
An attractor is a closed set of states A  S such that:
(a) for all s  A, ft(s)  A whenever t ≥ 0
(b) for some open U  A: for all s  U and open V  A, for some t, ft*(s)  V for all t* ≥ t
(c) if A*  A and A* satisfies (a) and (b), then A* = A
Informally, (a) means that once the system is in A, it stays in A; (b) means that A draws and keeps all
trajectories that ever come sufficiently close to it arbitrarily close to it (A attracts such trajectories);
(c) means that A is minimal in these respects.
The basin of attraction of A is the largest set U satisfying (b) (such a set is bound to exist).
A strange attractor is an attractor that exhibits sensitive dependence on initial conditions. Such
attractors are often fractal sets. They are important for the theory of chaos.
NB: The definition freely quantifies over the analogues of possible worlds (with the variable “s”) and
of sets of possible worlds (with the variables “U” and “V”). Without this it makes no sense.
The internal structure of states
Elements of S are typically interpreted as assignments of values to one or more variables, whose
number is the dimension of S. The variables may be global features of the underlying physical reality,
e.g. number of predators and number of prey. Mathematically, each state in a typical n-dimensional
system is just an n-tuple of numbers (or something more complicated if e.g. there is a temperature
variable for each point in the continuum of physical space).
In agent-based dynamical systems, each variable measures a feature of a given individual. If there
are m individuals, each with n relevant features, then the system is m.n-dimensional. In systems with
creation or extinction of individuals (particles, cells, organisms [compare J. Gunawardena, “Models
in systems biology: the parameter problem and the meanings of robustness”, in H. Lodhi, S.
Muggleton (eds.), Elements of Computational Systems Biology (Wiley, 2009), at p. 6]) it may be
contingent what individuals there are, e.g. a given individual may occur on some trajectories but not
on others. Mathematically, this is best handled by having variables for each individual that can occur
in some state or other. Thus, on the intended application, the identity of the states in effect encodes
the identities of all possible individuals. To quantify over states and sets of states as above is in effect
to quantify over possibilia (this is so even though the theory is general over many dynamical systems
of a given sort). This presents a major problem for contingentists, for whom it is contingent what
there is, so such quantification makes no obvious sense, but not for necessitists, for whom it is noncontingent what there is. Thorough-going contingentists can simulate necessitist quantification in
first-order logic but not in second-order logic; see T. Williamson, Modal Logic as Metaphysics,
chapter 7. The quantification over sets of states in the definition of ‘attractor’ corresponds to such
second-order quantification. Contingentists can’t just help themselves to it.
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Moral: The non-modal content of mathematics does not mean that its applications in natural science
are non-modal. Rather, it manifests the necessitist presuppositions of the scientific applications.
Contingentists who want to reconcile their modal metaphysics with scientific practice face a major
reconstructive challenge, to vindicate the free scientific use of quantification over states, sets of
states, & etc. There is no reason to expect that they can meet the challenge.
Metaphysical versus nomic modality
In the first instance, natural science seems to concern the domain of nomic possibility ◊N, rather
than the possibly much wider domain of metaphysical possibility ◊ (e.g. worlds with alternative laws
of nature). How much can we learn from natural science about metaphysical modality?
One strategy is to go with the idea proposed by Kripke in Naming and Necessity and developed by
Alexander Bird in Nature’s Metaphysics: Laws and Properties that nomic necessity is much closer to
metaphysical necessity than is usually supposed, perhaps even equivalent to it. Although there is
some mileage in that strategy, it will not be pursued here. Instead, we make some points consistent
with the assumption that nomic possibility is much narrower than metaphysical possibility.
Since nomic possibility implies metaphysical possibility (◊N… entails ◊…), any positive results of
natural science to the effect that something is nomically possible also imply that it is metaphysically
possible. Successful applied theories e.g. of dynamical systems have such positive implications.
Curtailing the state space typically disrupts its geometrical or topological structure and thereby
reduces the abductive power of the theory.
The analogies between metaphysical and nomic modality (e.g. both are alethic and closely related to
subjunctive conditionals) mean that if a structural principle holds for nomic modality, that is some
evidence that it also holds for metaphysical modality. Example: all the principles of the system T of
normal propositional modal logic.
In some cases, the salient purported counterexamples to a principle about metaphysical modality
arise within the realm of nomic modality. Thus, if they fail within that realm, they fail altogether.
Example: The (metaphysical) necessitist thesis is □  x □  y x=y. A typical purported contingentist
counterexample is that I would never have existed (been something) if my parents had never met.
Presumably, however, according to the contingentist, it is nomically as well as metaphysically
possible for my parents never to have met. Thus scientific evidence for the nomic necessitist thesis
□N  x □N  y x=y is evidence that the corresponding purported contingentist counterexample to
nomic necessitism fails, e.g. because if my parents had never met I would still have been something,
a merely possible person, in which case it also fails as a counterexample to metaphysical
necessitism. Even though nomic necessitism does not automatically yield metaphysical necessitism,
by defeating the objections to the former one also defeats the most threatening objections to the
latter.
Thus the distinction between nomic and metaphysical modality, though granted, makes less
difference than might have been supposed to the modal upshot of natural science.
A more general moral: the epistemology of modality cannot be treated in isolation, since our
scientific knowledge of modality is so tightly integrated with the rest of our knowledge. Not very
surprisingly, the abductive methodology of natural science is appropriate for modal inquiry.